Clockfrom a stick's shadow to pulsars

The first clocks: shadow

The simplest clock is an upright stick — a gnomon: its shadow turns and shortens over the day. The Egyptians already had L-shaped shadow clocks. The oldest surviving one is usually dated to the reign of Thutmose III (c. 1450 BC), though dates vary.

Obelisks may have served as huge gnomons. What is certain is that in 10 BC Emperor Augustus set up an Egyptian obelisk in Rome as the pointer of a great clock-calendar (the Horologium Augusti). The Greeks and Romans also used the hemicycle — a bowl carved into stone (Vitruvius credits it to Berossus).

Unequal hours

In antiquity the day — from sunrise to sunset — was divided into 12 hours, and so was the night. A daytime hour was therefore long in summer and short in winter. In Rome it lasted about 75 min in summer and about 45 min in winter; in Warsaw it would be about 83 and 37 min.

Equal hours caught on only with mechanical clocks (14th century). On the dial next to this you can see the length of an “ancient” hour for the chosen day.

Why is the blade tilted?

In a good sundial the blade (the style) is parallel to Earth's axis — it points at the celestial pole, near Polaris. The Sun, appearing to circle that axis at 15° an hour, then always casts the blade's shadow on the same line at the same hour — whatever the season.

blade angle to the horizontal = φ (latitude)

On a horizontal dial the lines are not 15° apart; instead tan θ = sin φ · tan(15° · t). For Warsaw the 1 h line is about 11.95° from the noon line.

4 minutes per degree

Earth turns 360° in 24 h, i.e. 1° every 4 minutes. In Poland the 15° E meridian runs through Zgorzelec and Stargard — there mean solar noon falls at about 12:00 CET. Warsaw (21° E) has it at 11:36 CET, Terespol at about 11:25. Between Poland's western and eastern edges that is about 40 minutes.

Spain uses Central European Time even though Madrid lies at 3.7° W. There the Sun culminates on average at 13:15 CET, and in midsummer at about 14:13–14:21 CEST.

Equation of time

Even in Zgorzelec the Sun does not always culminate at 12:00. True solar time “drifts” over the year from about −14 min (mid-February) to about +16 min (early November). This is the equation of time. It has two causes: Earth's orbit is elliptical (Earth moves faster in winter), and Earth's axis is tilted by about 23.4°.

If you photograph the Sun at the same clock time every day, its positions trace a figure eight — the analemma.

Time zones

In the past every town kept its own solar time. Trouble came with the railways: a timetable with a dozen local times was a nightmare. British railways adopted Greenwich time from the 1840s, and in 1883 the railroads of the USA and Canada introduced time zones. In 1884 a conference in Washington chose the Greenwich meridian as the prime meridian.

On Polish lands under Prussian rule, Central European Time applied from 1893; elsewhere local and railway times were used, among others (Warsaw time was about 1 h 24 min ahead of Greenwich). A single Central European Time for all of Poland was introduced in 1922.

Clepsydra = “water thief”

The Greek word klepsydra literally means “water stealer.” One of the oldest known water clocks was found at Karnak; it dates from the time of Amenhotep III (c. 1400 BC). It is a stone vessel with a hole near the bottom and sloping sides. Inside it has 12 columns of markings, one per month, because night hours changed length with the seasons.

Torricelli's law

Water flows out of the hole faster the higher the column above it:

v = 2gh

So in a cylinder the level falls more and more slowly — along a parabola h(t) = h₀(1 − t/T)². For it to fall evenly, the cross-section must grow like √h, i.e. the radius like h^1/4. The sloping walls at Karnak partly compensate for this.

Ctesibius and the first regulator

In the 3rd century BC Ctesibius of Alexandria built a clock in which water flows into a vessel from a tank kept at a constant level (the excess drains through an overflow). A float raises a pointer, and according to Vitruvius a rack could also drive rotating dials. The float valve is considered one of the oldest known feedback regulators.

A bridge to mechanics: Su Song

In China in 1088, Su Song's astronomical clock tower was completed (described in a treatise from 1092). A water wheel with scoops turned in steps: each scoop had to fill before the mechanism let the wheel move on. It is a kind of escapement — like those in later mechanical clocks, only driven by water.

The sand hourglass

It appears in Europe in the 14th century; the oldest reliable depiction is Ambrogio Lorenzetti's fresco in Siena (1338). Sand flows at an almost constant rate, nearly regardless of how much is left. The walls carry part of the grains' weight (the Janssen effect), and the flow is set mainly by the width of the neck (Beverloo's law).

On ships the sandglass measured the watches, and the bell was struck every half hour. Its running is far less affected by the ship's rolling than water would be.

Candles and incense

According to his biographer Asser, King Alfred the Great (9th century) measured time with candles divided into segments. In China, and later in Japan, incense clocks were used: a smoldering trail of incense marked the passing hours, sometimes even dropping a ball onto a gong.

Model: outflow without viscosity or surface tension, outlet small compared with the vessel. Sand — constant flow rate.

Power source

A weight of mass m that descends by a height h gives up potential energy:

E = m · g · h = 4 kg · 9.81 ms² · 1.2 m ≈ 47 J

Spread over 8 days (691,200 s), that is a power of just about 68 µW — thousands of times less than an LED. A mainspring in a barrel works the same way, except that the energy is stored in a coiled steel ribbon.

Gear train

When a wheel with z₁ teeth turns a pinion with z₂ teeth, the pinion spins z₁/z₂ times faster. In our clock:

968 · 608 · 567 = 12 · 7.5 · 8 = 720

The barrel turns once in 12 h, so the escape wheel makes 720 turns in 12 h, i.e. 1 turn per minute. The center wheel turns 60 times slower: once an hour — the minute hand sits on its arbor.

Escapement

The anchor won't let the escape wheel spin freely: at every swing it lets the wheel advance by half a tooth. Along the way a tooth pushes on the slanted face of a pallet (the impulse) and gives the pendulum back the energy it loses to friction.

30 teeth · 2 s = 60 s

That's why the seconds hand of a pendulum clock sits directly on the escape-wheel arbor and jumps once a second — after every “tick” and every “tock.”

Pendulum

The period of small swings depends only on the length l and the gravitational acceleration g (we take 9.81 m/s²):

T = 2π lg  ⇒  l = gπ² ≈ 0.994 m for T = 2 s

The daily error is Δ = 86,400 s · (T₀/T − 1). For a seconds pendulum every millimeter of length is worth about 43 s a day.

Motion works

The hour hand must turn 12 times slower than the minute hand. A small gear train under the dial does this: the cannon pinion (10 t), the minute wheel (30 t) with its pinion (8 t), and the hour wheel (32 t):

1030 · 832 = 13 · 14 = 112

The cannon pinion is friction-fitted onto the center-wheel arbor — which is why you can reset the hands without damaging the mechanism.

Balance and hairspring

A pendulum wouldn't work in a watch (the watch moves around), so it is replaced by a balance wheel and a spiral spring — the hairspring:

T = 2π Iκ

I — moment of inertia of the balance, κ — stiffness of the hairspring. Our balance oscillates at 2.5 Hz (18,000 vibrations per hour). The escape wheel (15 t) turns once every 15 · 0.4 s = 6 s, and the fourth wheel (80/8 = 10 times slower) once a minute.

Time is longitude

Earth turns 15° an hour. If at local noon (Sun at its highest) a clock set in the home port shows 17:07, the ship is 5 h 7 min west of the port, i.e. at about 76.8° W — almost exactly off the coast of Jamaica.

Δλ = 15° · Δt [h]

In practice you also have to account for the equation of time — true solar noon drifts by ±16 min over the year.

1 second = 464 meters

A day has 86,400 s, and the equator is about 40,075 km around. One second of clock error is 15″ of longitude, i.e. about 463.8 m at the equator (less away from it: times cos φ).

1 s → 15″ → 40,075 km86,400 ≈ 464 m

A clock that gains 1 minute a day puts you about 830 km off at the equator after a month.

Scilly 1707 and the 1714 prize

In 1707 Admiral Cloudesley Shovell's squadron was wrecked on the rocks of the Isles of Scilly — more than 1,400 sailors died (estimates reach 2,000). The cause was navigation errors, including an uncertain position.

In 1714 the British Longitude Act promised £20,000 for a method of finding longitude to within half a degree (about 30 nautical miles at the equator) on a voyage to the West Indies. Half a degree is 2 minutes of time, so on a 6-week voyage the clock could be off by at most about 2.86 s a day.

H4 in practice

John Harrison, a self-taught carpenter from Yorkshire, worked on the problem for over 30 years. On the trial voyage to Jamaica (1761–1762), H4 — after allowing for its declared rate — lost only about 5 s by the most commonly quoted reckoning. That is about 1.25′ of longitude, or little more than 1 nautical mile.

Harrison received full recognition (and an additional award from Parliament) only in 1773. You'll find the chronometer's mechanism in the Mechanical clock mode.

The competition: the Moon

The lunar distance method: the Moon moves against the stars like the hand of a giant clock. From the angle between the Moon and a star, plus tables (from 1767 in the Nautical Almanac), navigators worked out Greenwich time. It was cheaper than a chronometer, but required long calculations.

Model: the ship sails a fixed route, the chronometer has a constant unforeseen rate, and the map is illustrative (equirectangular projection). You can read about orbits and gravity on the Gravity page.

Piezoelectricity

A quartz crystal that is squeezed or bent produces an electric voltage, and under a voltage it deforms itself. The brothers Pierre and Jacques Curie discovered this in 1880. Cut into the shape of a tiny tuning fork, quartz rings in an electronic circuit like a bell, always at the same frequency.

Why exactly 32,768?

32,768 = 2¹⁵. A flip-flop is a simple circuit that changes state on every second pulse, so it divides the frequency by 2. Fifteen of them in a row give exactly 1 pulse per second for the stepper motor that drives the hands.

The frequency is also above the range of hearing, and the tuning fork stays small and power-efficient.

From the lab to the wrist

The first quartz clock was built by Warren Marrison and J. W. Horton at Bell Labs in 1927 — a cabinet full of vacuum tubes. In December 1969 Seiko launched the Astron, the first quartz wristwatch. Today an ordinary quartz watch is typically off by about ±15 s a month.

The enemy: temperature

The tuning fork vibrates fastest (and most accurately) at about 25 °C; both cooling and warming slow it down — along a parabola. In freezing weather at 0 °C a watch would lose about 1.8 s a day. On the wrist it usually sits at about 30 °C, so the deviation is small.

We show the tuning fork's vibrations and the LED states in slow motion; in a real watch the divider is a tiny part of a single chip.

The feedback loop

A quartz oscillator generates microwaves at a frequency close to 9,192,631,770 Hz. Cesium atoms fly through a cavity; only at exactly the right frequency do most of them change state. A magnet steers those atoms into a detector. The electronics “wobble” the frequency up and down and shift it to where the signal is strongest.

Essen, Parry and fountains

In 1955 Louis Essen and Jack Parry built the first accurate cesium clock at Britain's National Physical Laboratory. In the 1990s cesium fountains appeared: laser-cooled atoms are tossed upward and fall back through the cavity twice. The longer measurement time (about 1 s) means a narrower resonance and an accuracy of order 10⁻¹⁵–10⁻¹⁶.

Optical clocks

Clocks based on light oscillations (e.g. strontium atoms in an optical lattice) reach about 10⁻¹⁸. They are so sensitive that they feel gravity: a clock raised by 1 cm runs faster by about 1.1 · 10⁻¹⁸. In 2010 NIST measured the rate difference between clocks about 33 cm apart vertically. In the future (perhaps around 2030) the second is expected to be redefined using optical clocks.

Time in Poland PL

Official time in Poland is based on the UTC(PL) scale, maintained with atomic clocks by the Central Office of Measures (Główny Urząd Miar) in Warsaw; the Astrogeodynamic Observatory of the Space Research Center (CBK PAN) in Borówiec also contributes to world atomic time, among others. Most “radio-controlled” clocks in Poland set themselves by the DCF77 signal — a 77.5 kHz longwave transmitter in Mainflingen near Frankfurt.

Leap seconds

Atomic time (TAI) runs evenly, but Earth rotates unevenly and is slowly braking. So universal time UTC was kept close to Earth's rotation by adding leap seconds from 1972 on — 27 so far, the last on December 31, 2016. Today TAI − UTC = 37 s. In 2022 the General Conference on Weights and Measures decided that by 2035 at the latest the allowed difference will be increased — in practice, the end of leap seconds.

“Little green men”

In 1967 the doctoral student Jocelyn Bell (now Bell Burnell), working with Antony Hewish in Cambridge, noticed regular pulses every 1.337 s in a radio telescope's recording. The source was jokingly named LGM-1 (little green men). It soon turned out to be a neutron star.

The 1974 Nobel Prize in Physics went to Antony Hewish and Martin Ryle; Jocelyn Bell Burnell was not included, which is still debated today.

A star the size of a city

A neutron star has a mass of about 1.4 solar masses but a diameter of only about 20 km. A teaspoon of its matter would weigh billions of tonnes. Its magnetic field is enormous, and the beams come out of the magnetic poles, which do not line up with the rotation axis — hence the lighthouse effect. How such stars form you'll see on the Life of stars page.

Nature's clocks

Millisecond pulsars such as PSR B1937+21 have been “spun up” by matter from a companion star. Over years, some rival atomic clocks in stability. In 2023 networks monitoring dozens of such pulsars (NANOGrav, the European EPTA, the Australian PPTA and others) announced evidence of a gravitational-wave background, which very slightly disturbs the pulse arrival times. More about extreme gravity on the Black hole page.

Earth's address in pulsars

The plaques on the Pioneer 10 and 11 probes and the covers of the Voyagers' Golden Records carry a map of 14 pulsars: lines from the Sun to the pulsars, with their periods written in binary, in units of the period of the 21 cm hydrogen line. Another civilization could read from it where and when the probe was sent.

Model: the beam is a cone, the magnetic axis is tilted 40° to the rotation axis, and Earth lies exactly in the beam's path. The display speed can be slowed down; the periods are real.